You should see iPhone changes on desktop and vice versa if 'Sync Automatically' is enabled in Notezilla . Just check it once. Sync happens automatically every minute on iPhone & desktop computers.
However, if iPhone is not showing latest notes automatically, that's because "Push Notifications" was disabled when the app was first installed.
Go to Notification Center on your iPhone Settings, tap on Notezilla and enable all notifications (Sound etc). This should enable push notifications. After you enable, restart your phone once and check if automatic sync works.